KPZ relation does not hold for the level lines and flow lines of the Gaussian free field

• 成果类型: Article
• 署名作者: Aru, Juhan
• 署名单位: Ecole Normale Superieure de Lyon (ENS de LYON)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0597-1
• 发表日期: 2015
• 页码: 465-526
• 关键词: random-walks; QUANTUM-GRAVITY; sle

摘要

In this paper we mingle the Gaussian free field, the Schramm-Loewner evolution (SLE) and the KPZ relation in a natural way, shedding new light on all of them. In particular, we describe the quantum fractal behaviour of the level lines and the flow lines of the Gaussian free field by determining their quantum Minkowski dimensions. As a corollary we deduce that the usual KPZ relation is not satisfied. In order to determine the fractal dimensions, we have to make a technical detour: by a careful study of a certain diffusion process, we provide exact estimates of the exponential moments of winding of chordal SLE curves conditioned to pass nearby a fixed point. This extends previous results on winding of SLE curves by Schramm.